In mathematics, a Grassmannian, also known as a Grassmann manifold, is a differentiable manifold-dimensional linear subspaces-dimensional vector space over a field that has a differentiable structure, so it is the same as the projective space. is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a nonsingular projective algebraic variety. The Grassmannian is named for the German polymath, linguist and mathematician Hermann Grassmann, who introduced the concept to mathematics.

History

The earliest work on a non-trivial Grassmannian was by Julius Plücker, parameterizing them by what are now called Plücker coordinates. Hermann Grassmann later generalized the concept.

.

Motivation

By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differentiable manifold enables smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space, the tangent space. The map its tangent space defines a map from -dimensional vector subspace. This idea is similar to the Gauss map for surfaces in a 3-dimensional space.)

This can be extended to all vector bundles to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. The properties of vector bundles are thus related to the properties of the corresponding maps. In particular vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.

Low dimensions

For , the Grassmannian is the space of lines through the origin in -space, so it is the same as the projective space of dimensions.

For , the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the single line through the origin that is perpendicular to that plane (and vice versa); hence the spaces , , and (the projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is .

Differentiable manifold

with the structure of a differentiable manifold, a basis linearly independent. The homogeneous coordinates maximal rank rectangular matrix if and only if

of the general linear group of invertible. This defines an equivalence relation.

, elementary column operations) to obtain its reduced column echelon form are linearly independent, the result has the form

is nonsingular. Column operations can reduce this submatrix to the identity matrix. This gives the following definition:

representing may take arbitrary values, and they define a diffeomorphism matrices. This can be denoted by

are related by the transition relations

are invertible. This may equivalently be written as

as a differentiable manifold and also as an algebraic variety.

Orthogonal projections

An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators. For this, a positive definite real or Hermitian inner product whose image into the orthogonal direct sum

and defining

as its image. Since the rank of an orthogonal projection operator equals its trace:

, respectively.

the Grassmannian is a compact Hausdorff into a metric space with metric

-dimensional subspaces, where denotes the operator norm, and hence an equivalent metric.

For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.

Affine algebraic varieties

that satisfy the three conditions:

• is a projection operator.

• is symmetric.

• .

to the projection matrix

component column vectors.

satisfying

• is a projection operator.

• is self-adjoint.

• ,

where the self-adjointness is

Homogeneous space

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space acts transitively can be expressed as

of the stabilizer:

-action.

can be identified with the quotient space

of left cosets.

is considered as a Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a ground field is an algebraic group, and this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete is a parabolic subgroup.

. The real orthogonal group is

1. ,

. This gives an identification as the homogeneous space

1. .

components) gives the isomorphism

Over let the unitary group act transitively, and analogously

,

In particular, this shows that the Grassmannian is compact, and of (real or complex) dimension .

Scheme

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.

Representable functor

is a quasi-coherent sheaf, the Grassmannian functor associates the set of quotient modules of

.

This functor. The latter is projective and the usual Grassmannian variety of the dual space, giving the canonical isomorphism

over the residue field.

Universal family

. The quotient homomorphism induces a closed immersion from the projective bundle:

For any morphism of -schemes:

this closed immersion induces a closed immersion

Conversely, any such closed immersion comes from a surjective to a locally free module.

in

is the set

Plücker embedding

The Plücker embeddingth exterior power.

denotes the projective equivalence class.

gives a different wedge product, but the two differ only by a non-zero scalar multiple (the determinant of the change of basis as the span such that

1. .

Plücker coordinates and Plücker relations

The Plücker embedding of the Grassmannian satisfies simple quadratic relations called the Plücker relations-component column vectors forming the transpose of the corresponding homogeneous coordinate matrix:

.

positive integers, respectively, the following homogeneous quadratic equations, known as the Plücker relations, or the Plücker-Grassmann relations under the Plücker map embedding:

omitted. These are consistent, determining a nonsingular projective algebraic variety.

, this single Plücker relation is

under the Plücker embedding.

Duality

-dimensional quotient space. This gives the natural short exact sequence:

Taking the dual to each of these three spaces and the dual linear transformations

. In terms of the Grassmannian, this gives a canonical isomorphism

-dimensional orthogonal complement.

Schubert cells

The detailed study of Grassmannians makes use of a decomposition into affine subpaces called Schubert cells, which were first applied in enumerative geometry are defined in terms of a specified complete flag. For any integer partition

of weight

consisting of weakly decreasing non-negative integers

whose Young diagram whose intersections have the following dimensions

These are affine spaces, and their closures (within the Zariski topology) are Schubert varieties.

As an example of the technique, determining the Euler characteristic of -dimensional subspaces of into -dimensional subspaces of that contain . This gives recursive formulae:

Solving these recursion relations is even is odd and

otherwise.

Cohomology ring

-space. Mapping each point in a k-plane to the point representing that plane in the Grassmannian yields the vector bundle, which generalizes the tautological bundle of a projective space. The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes. In particular, all of the integral cohomology is at even degree as in the case of a projective space.

. Then the relations state that the direct sum is trivial. Functoriality of the total Chern classes shows that this relation is

The quantum cohomology ring was calculated by Edward Witten. The generators are identical to those of the classical cohomology ring, but with the top relation

reflecting the existence in the corresponding quantum field theory of an instanton fermionic zero-modes units.

Associated measure

-dimensional Euclidean space is the unit Haar measure on the orthogonal group ,

; that is,

is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.

Oriented Grassmannian

.

As a homogeneous space it can be expressed as:

Orthogonal isotropic Grassmannians

Given a real or complex nondegenerate symmetric bilinear form for which

i.e., totally isotropic subspaces. Maximal isotropic Grassmannians with respect to a real or complex scalar product are related to Cartan's theory of spinors. Under the Cartan embedding, their connected components are equivariantly diffeomorphic to the projectivized minimal spinor orbit. Under the spin representation, the projective pure spinor variety which, similarly to the image of the Plücker map embedding, is cut out as the intersection of a number of quadrics, becomes Cartan quadrics.

Applications

Pure mathematics

A key application of Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.

Another important application is Schubert calculus, which is the enumerative geometry involved in calculating the number of points, lines, planes, etc. in a projective space that intersect a given set of points, lines, etc., using the intersection theory of Schubert varieties. Subvarieties of Schubert cells can also be used to parametrize simultaneous eigenvectors of complete sets of commuting operators in quantum integrable spin systems, such as the Gaudin model, using the Bethe ansatz method.

Physics

A further application is to the solution of hierarchies of classical completely integrable systems of partial differential equations, such as the Kadomtsev–Petviashvili equation and the associated KP hierarchy. These can be expressed in terms of Abelian group flows on an infinite-dimensional Grassmann manifold. The KP equations, expressed in Hirota bilinear form in terms of the KP Tau function are equivalent to the Plücker relations. A similar construction holds for solutions of the BKP integrable hierarchy, in terms of Abelian group flows on an infinite dimensional, maximal isotropic, Grassmann manifold.

Finite dimensional positive Grassmann manifolds can be used to express soliton solutions of KP equations that are nonsingular for real values of the KP flow parameters.

The scattering amplitudes of subatomic particles in maximally supersymmetric super Yang-Mills theory may be calculated in the planar limit via a positive Grassmannian construct called the amplituhedron.

In quantum chemistry and many-body quantum physics, the Grassmannian can be identified with the space of pure many-particle fermionic states which can be written as a single Slater determinant. The geometric properties of the Grassmannian can be used to study theoretical properties of the Hartree-Fock method.

Computer science

Grassmann manifolds have found applications in computer vision tasks of video-based face recognition and shape recognition, and are used in the data-visualization technique known as the grand tour.

Grassmann flows are under consideration as a potential replacement for attention mechanisms in large language models, based on research showing competitive performance with better computational efficiency and interpretability. Using low-dimensional subspaces on the Grassmann manifold, causal Grassmann layers allow:

• reducing token states to a lower dimension (e.g., from 256 to 32).

• encoding local token pairs as 2D subspaces using Plücker coordinates.

• fusing into hidden states through gated mixing, allowing information to "flow" via controlled deformations over multi-scale local windows.

• avoiding high-dimensional, pairwise tensor operations of standard self-attention, confining computations to a structured manifold for more traceable geometry.

In contrast to attention models, which process context at the scale of (O(n²)), Grassmann flows address scale linearly (O(n) for fixed subspace rank), enabling efficiency on long sequences without approximations like sparse attention.

See also

• Schubert calculus

• For an example of the use of Grassmannians in differential geometry, see Gauss map

• In projective geometry, see Plücker embedding and Plücker co-ordinates.

• Flag manifolds are generalizations of Grassmannians whose elements, viewed geometrically, are nested sequences of subspaces of specified dimensions.

• Stiefel manifolds are bundles of orthonormal frames over Grassmannians.

• Given a distinguished class of subspaces, one can define Grassmannians of these subspaces, such as Isotropic Grassmannians or Lagrangian Grassmannians .

• Isotropic Grassmannian

• Lagrangian Grassmannian

• Grassmannians provide classifying spaces in K-theory, notably the classifying space for U(n). In the homotopy theory of schemes, the Grassmannian plays a similar role for algebraic K-theory.

• Affine Grassmannian

• Grassmann bundle

• Grassmann graph

References

Sources

• section 1.2

• see chapters 5–7